1. The Three Common Approaches for Calculating Value at Risk
CHAPTER 3
The Three Common Approaches for Calculating Value at Risk

2. INTRODUCTION VaR is a good measure of risk
To estimate the value's probability distribution, we use two sets of information
the current position, or holdings, in the bank's trading portfolio
an estimate of the probability distribution of the price changes over the next day.

3. INTRODUCTION
The estimate of the probability distribution of the price changes is based on the distribution of price changes over the last few weeks or months.
The goal of this chapter is to explain how to calculate VaR using the three methods that are in common use:
Parametric VaR
Historical Simulation
Monte Carlo Simulation.

4. LIMITATIONS SHARED BY ALL THREE METHODS
It is important to note that while the three calculation methods differ, they do share common attributes and limitations.
Each approach uses market-risk factors
Risk factors are fundamental market rates that can be derived from the prices of securities being traded
Typically, the main risk factors used are interest rates, foreign exchange rates, equity indices, commodity prices, forward prices, and implied volatilities
By observing this small number of risk factors, we are able to calculate the price of all the thousands of different securities held by the bank

5. LIMITATIONS SHARED BY ALL THREE METHODS
Each approach uses the distribution of historical price changes to estimate the probability distributions.
This requires a choice of historical horizon for the market data
how far back should we go in using historical data to calculate standard deviations?
This is a trade-off between having large amounts of information or fresh information

6. LIMITATIONS SHARED BY ALL THREE METHODS
Because VaR attempts to predict the future probability distribution, it should use the latest market data with the latest market structure and sentiment
However, with a limited amount of data, the estimates become less accurate
There is less chance of having data that contains those extreme, rare market movements which are the ones that cause the greatest losses

7. LIMITATIONS SHARED BY ALL THREE METHODS
Each approach has the disadvantage of assuming that past relationships between the risk factors will be repeated
it assumes that factors that have tended to move together in the past will move together in the future

8. LIMITATIONS SHARED BY ALL THREE METHODS
Each approach has strengths and weaknesses when compared to the others, as summarized in Figure 6-1
The degree to which the circles are shaded corresponds to the strength of the approach
The factors evaluated in the table are
the speed of computation
the ability to capture nonlinearity
Nonlinearity refers, to the price change not being at linear function of the change in the risk factors. This is especially important for options
the ability to capture non-Normality
non-Normality refers to the ability to calculate the potential changes in risk factors without assuming that they have a Normal distribution
the independence from historical data

9. LIMITATIONS SHARED BY ALL THREE METHODS

10. PARAMETRIC VAR
Parametric VaR is also known as Linear VaR, Variance-Covariance VaR
The approach is parametric in that it assumes that the probability distribution is Normal and then requires calculation of the variance and covariance parameters.
The approach is linear in that changes in instrument values are assumed to be linear with respect to changes in risk factors.
For example, for bonds the sensitivity is described by duration, and for options it is described by the Greeks

11. PARAMETRIC VAR The overall Parametric VaR approach is as follows:
Define the set of risk factors that will be sufficient to calculate the value of the bank's portfolio
Find the sensitivity of each instrument in the portfolio to each risk factor
Get historical data on the risk factors to calculate the standard deviation of the changes and the correlations between them
Estimate the standard deviation of the value of the portfolio by multiplying the sensitivities by the standard deviations, taking into account all correlations
Finally, assume that the loss distribution is Normally distributed, and therefore approximate the 99% VaR as 2.32 times the standard deviation of the value of the portfolio

12. PARAMETRIC VAR Parametric VaR has two advantages:
It is typically 100 to 1000 times faster to calculate Parametric VaR compared with Monte Carlo or Historical Simulation.
Parametric VaR allows the calculation of VaR contribution, as explained in the next chapter.

13. PARAMETRIC VAR Parametric VaR also has significant limitations:
It gives a poor description of nonlinear risks
It gives a poor description of extreme tail events, such as crises, because it assumes that the risk factors have a Normal distribution. In reality, as we found in the statistics chapter, the risk-factor distributions have a high kurtosis with more extreme events than would be predicted by the Normal distribution.
Parametric VaR uses a covariance matrix, and this implicitly assumes that the correlations between risk factors is stable and constant over time

14. How to calculate volatility of each asset?
JP Morgan's RiskMetrics system
Equally-Weighted Moving Average (that is Simple Moving Average (SMA); standard deviation)
Exponentially-Weighted Moving Average (EWMA)
λ: decay rate, 0<λ<1. The more the λ value, the less last observation affects the current dispersion estimation.
The formula of the EWMA model can be rearranged to the following form:

15. How to calculate volatility of each asset?
The EWMA model has an advantage in comparison with SMA, because the EWMA has a memory.
Using the EWMA allows one to capture the dynamic features of volatility. This model uses the latest observations with the highest weights in the volatility estimate. However, SMA has the same weights for any observation.
JP Morgan suggests: the optimal value for current daily dispersion (volatility) is =0.94; the optimal value for current monthly dispersion (volatility) is =0.97

16. Example: One Asset
The first example calculates the stand-alone VaR for a bank holding a long position in an equity. The stand-alone VaR is the VaR for the position on its own without considering correlation and diversification effects from other positions
The present value of the position is simply the number of shares (N) times the value per share, (Vs)
PV$ = N * Vs

17. Example: One Asset
The change in the value of the position is simply the number of shares multiplied by the change in the value of each share:
ΔPV$ = N * ΔVs
The standard deviation of the value is the number of shares multiplied by the standard deviation of the value of each share
σv = N * σs
we have assumed that the value changes are Normally distributed, there will be a 1chance that the loss is more than 2.32 standard deviations; therefore, we can calculate the 99 VaR as follows
VaR = 2.32 * N *σs

18. Example: Two Assets Portfolio: P=A1 (amount: w1)+A2 (amount: w2)
Portfolio variance
Portfolio’s VaR
VaR1 and VaR2 is the single asset’s VaR:
VaR1= and VaR2=

19. Example: n Assets
If the number of asset is n, the portfolio’s variance
The portfolio VaR is
Hence, the portfolio VaR depends on the each asset’s variance and the correlation coefficient between the returns of each asset.

20. Example

21. HISTORICAL-SIMULATION VAR
Conceptually, historical simulation is the most simple VaR technique, but it takes significantly more time to run than parametric VaR.
The historical-simulation approach takes the market data for the last 250 days and calculates the percent change for each risk factor on each day
Each percentage change is then multiplied by today's market values to present 250 scenarios for tomorrow's values.
For each of these scenarios, the portfolio is valued using full, nonlinear pricing models. The third-worst day is the selected as being the 99% VaR.

22. How to calculate historical simulation VaR
Step 1: download the historical data of asset price (last 250 days)
Step 2: calculate the change rate (returns) of asset
Step 3: sort the returns of asset from the lowest to highest
Step 4: given the significant level to find the historical simulation VaR
Ex: 1001 price observations (1000 returns observations), significant level=1%, then 1000*1%=10. That is find the historical simulation VaR is the tenth returns

23. HISTORICAL-SIMULATION VAR
There are two main advantages of using historical simulation:
It is easy to communicate the results throughout the organization because the concepts are easily explained
There is no need to assume that the changes in the risk factors have a structured parametric probability distribution
no need to assume they are Normal with stable correlation

24. HISTORICAL-SIMULATION VAR
The disadvantages are due to using the historical data in such a raw form:
The result is often dominated by a single, recent, specific crisis, and it is very difficult to test other assumptions.
The effect of this is that Historical VaR is strongly backward-looking, meaning the bank is, in effect, protecting itself from the last crisis, but not necessarily preparing itself for the next

25. HISTORICAL-SIMULATION VAR
There can also be an unpleasant "window effect."
When 250 days have passed since the crisis, the crisis observation drops out of our window for historical data and the reported VaR suddenly drops from one day to the next.
This often causes traders to mistrust the VaR because they know there has been no significant change in the risk of the trading operation, and yet the quantification of risk has changed dramatically

26. MONTE CARLO SIMULATION VAR
Monte Carlo simulation is also known as Monte Carlo evaluation (MCE). It estimates VaR by randomly creating many scenarios for future rates using nonlinear pricing models to estimate the change in value for each scenario, and then calculating VaR according to the worst losses

27. MONTE CARLO SIMULATION VAR
Required:
(1) for each risk factor, specification of a stochastic process (i.e., distribution and parameters)
assumes that there is a known probability distribution for the risk factors.
the choice of distributions and parameters such as risk and correlations can be derived from historical data
(2) valuation models for all assets in the portfolio
fictitious price paths are simulated for all variables of interest.

28. MONTE CARLO SIMULATION VAR
Monte Carlo simulation has two significant advantages:
Unlike Parametric VaR, it uses full pricing models and can therefore capture the effects of nonlinearities
Unlike Historical VaR, it can generate an infinite number of scenarios and therefore test many possible future outcomes

29. MONTE CARLO SIMULATION VAR
Monte Carlo has two important disadvantages:
The calculation of Monte Carlo VaR can take 1000 times longer than Parametric VaR because the potential price of the portfolio has to be calculated thousands of times
The computation speed will lower.
Unlike Historical VaR, it typically requires the assumption that the risk factors have a known probability distribution .

30. MONTE CARLO SIMULATION VAR
The Monte Carlo approach assumes that there is a known probability distribution for the risk factors.
The usual implementation of Monte Carlo assumes a stable, Normal distribution for the risk factors.
This is the same assumption used for Parametric VaR.
The analysis calculates the covariance matrix for the risk factors in the same way as Parametric VaR

31. MONTE CARLO SIMULATION VAR
But unlike Parametric VaR
Decomposes the covariance matrix and ensures that the risk factors are correlated in each scenario
The scenarios start from today's market condition and go one day forward to give possible values at the end of the day
Full, nonlinear pricing models are then used to value the portfolio under each of the end-of-day scenarios.
For bonds, nonlinear pricing means using the bond-pricing formula rather than duration
for options, it means using a pricing formula such as Black-Scholes rather than just using the Greeks.